To prove Proposition REF , we will approximate the logarithm of \(\zeta (s)\)  by a truncated sum over primes \(\sum _{p\le X} p^{-s}\) .
Following the works [1]}, [2]}, [3]}, we will break up this sum into increments which have progressively smaller variance.
This in turn allows us to work with a Dirichlet polynomial of length \(T^\theta \)  for some small but fixed \(\theta > 0\) , which is long enough to give a good enough approximation of \(\zeta (s)\) .
